Contents

Idea

The notion of representation is closely related to, or even identical, to that of action: some object CC that has a notion of composition is represented on some object DD that has a notion of composition. In this generality representation is just another word for functor (or potentially ∞\infty-functor). But in practice the term representation is typically used in the context of representation theory, where we attempt to study CC in terms of its representations in DD, where DD is typically rather more familiar.

Most specifically, one studies representations of a group by linear endomorphisms of a vector space; that is, CC is (the delooping of) a group and DD is Vect. However, the typical tools in representation theory these days involve vast generalizations of the notion of a linear representation of a group; for instance, one studies D-modules on action groupoids G//AdGG //_{Ad}G and things like that. This may be thought of as studying representations with values in ∞-vector spaces.

Historical Idea (with pedagogic overtones)

Representing Groups

The notion of a group of ‘operators’ was already being used in about 1832 in work by Galois, and others, but there was not a definition of an abstract group until 40 years later when Cayley wrote:

A group is defined by the law of composition of its members.

(see the article on The abstract group concept in the St Andrews History of Mathematics archive.) Groups as well behaved sets of functions were beginning to be well understood and used, for instance in Klein’s work on geometry. Cayley proved that every finite group could be realised as a group of permutations. The theory of representations grew from that.

An abstract group could be studied by mapping it into a group of permutations or of invertible matrices, as then you could bring techniques from one area of mathematics (linear algebra) to the assistance of another, the Theory of Abstract Groups. This was exploited fully by Frobenius, Burnside, Schur and later Bauer.

That is one theme: you take an abstract algebraic thing and study it by mapping it into a similar structure which you think you know more about!

This also uses another basic idea from the start of group theory. The earlier pioneers thought of groups as groups of ‘operators’, but that means they have to operate on something. To make things more explicit, let GG be a finite group then we know there are homomorphisms from GG into SnS_n. (We could take nn to be the order of the group and use Cayley’s theorem but we do not assume that is the case.) If we look at such a homomorphism we get an action of GG on an nn-element set.

Similarly if we take a homomorphism from GG into a group of invertible matrices Gln(K)Gl_n(K), for KK a field, say, then we get a linear action of GG on the vector space, KnK^n.

As you would expect, we can generalise and categorify this basic idea in several useful ways.

We can think of GG as a groupoid, BG\mathbf{B}G, (the delooping of GG), and then a linear representation / action will be a functor from BG\mathbf{B}G to VectVect, the category of vector spaces over KK. We could replace GG by a general groupoid, or a general category, but then a representation of that is the same as a diagram of that ‘shape’ in VectVect. We could replace VectVect by another more general category, or higher category, but if we are thinking of diagrams as representations, perhaps we should not totally forget that the term ‘representation’ did mean a process whereby the perhaps abstract ‘syntactical’ objects of the category gain a ‘semantic’ meaning, as ‘operations’ of some type, and which in turn, can be usefully used to gain information on the inherent structure.

General definition

In a rather general form, we therefore have a representation of a category CC in a category DD is simply a functorF:C→DF\colon C \to D. Similarly, an homomorphism between representations (“intertwiner”) is simply a natural transformation between functors when they are being thought of as representations.

The term ‘representation’ is most often used when one or more of the following conditions apply:

DD is the category kk-Vect of vector spaces over some fieldkk; one then has a kk-linear representation.

CC is the delooping of a group; one then has a group representation in DD. Such a representation gives us a specific object VV of DD; we say that we have a representation of GG on VV.